Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are mutually orthogonal, for $k<n$? If so, might that conclusion be useful?

<b>PS:</b> Here's an example. This matrix is not symmetric but would become so if the $(1,2)$ entry were changed. Six of its seven eigenvectors are all orthogonal to each other.
$$
\left[
\begin{array}{rrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & -2 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & -2 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & -2 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & -2 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & -1
\end{array}
\right]
$$